Symmetric functions and regularity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:07:44Z http://mathoverflow.net/feeds/question/61720 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61720/symmetric-functions-and-regularity Symmetric functions and regularity Denis Serre 2011-04-14T16:20:42Z 2011-04-14T17:45:59Z <p>Let $f:\mathbb R^2\rightarrow\mathbb R$ be a symmetric function: $f(y,x)=f(x,y)$. It can therefore be written has a function of the elementary symmetric polynomials, here $f(x,y)=F(x+y,xy)$, where $F(\sigma,\mu)$ is defined over $4\mu\le\sigma^2$.</p> <p>Let me assume that $f$ is of class $\mathcal C^r$. It is clear from the IFT that $F$ is $\mathcal C^r$ too, away from the critical line $4\mu=\sigma^2$.</p> <blockquote> <p>What is the regularity of $F$ at the boundary ? My guess is $\mathcal C^{r/2}$ but I have been unable to prove it or to locate such a result.</p> </blockquote> <p>For instance, if $f(x,y)=|y-x|^r$ and $r>0$ is not an integer, then $F=|\sigma^2-4\mu|^{r/2}$ is only $\mathcal C^{r/2}$ and not more.</p> http://mathoverflow.net/questions/61720/symmetric-functions-and-regularity/61727#61727 Answer by Laurent Moret-Bailly for Symmetric functions and regularity Laurent Moret-Bailly 2011-04-14T17:45:59Z 2011-04-14T17:45:59Z <p>Just note that $F(\sigma,\mu)=f\left(\frac{\sigma+\sqrt{\sigma^2-4\mu}}{2}, \frac{\sigma-\sqrt{\sigma^2-4\mu}}{2}\right)$. </p>